Problem 66 - Algebra for College Students

Question

Use an algebraic approach to solve each problem. Barry sells bicycles on a salary-plus-commission basis. He receives a monthly salary of \(\$ 300\) and a commission of \(\$ 15\) for each bicycle that he sells. How many bicycles must he sell in a month to have a total monthly income of \(\$ 750\) ?

Step-by-Step Solution

Answer

Barry must sell 30 bicycles.

Step 1: Define the variables

Let \( x \) represent the number of bicycles Barry needs to sell in a month. We need to find \( x \) such that his total income is \( \$750 \).

Step 2: Express the income formula

Barry's total income comes from his monthly salary and his commission. His income formula is: \[ \, \text{Income} = \text{Salary} + \text{Commission} \, \] Using the information given, \[ \, \text{Income} = 300 + 15x \, \] where \( x \) is the number of bicycles sold.

Step 3: Set up the equation

We want Barry's total income to be \( \$750 \). Set the equation based on the income formula: \[ \, 300 + 15x = 750 \, \]

Step 4: Solve the equation for x

Subtract 300 from both sides to isolate the term with \( x \):\[ \, 15x = 750 - 300 \, \] Simplifying this gives: \[ \, 15x = 450 \, \]

Step 5: Find x by dividing both sides by 15

Divide both sides of the equation by 15 to solve for \( x \):\[ \, x = \frac{450}{15} \, \] Simplify this to find \( x = 30 \).

Key Concepts

Equations

In algebra, equations are essential as they help us model real-world problems. An equation is a mathematical statement that asserts the equality of two expressions. In Barry's case, we used an equation to find out how many bicycles he needs to sell to reach a desired income. This equation takes the form: \( 300 + 15x = 750 \), where each part of the equation reflects a component of his income.
This simple linear equation consists of two main elements:

• A constant, which is Barry's fixed monthly salary of \(\\(300\).
• A variable term \( 15x \), representing the commission per bicycle sold.

The sum of these two terms is set equal to Barry's desired income, \(\\)750\). This equation essentially tells us that Barry's salary plus his commission must total his goal income.

Variables

Variables are symbols used in algebra to represent unknown values or quantities that can change. In our problem, the variable \( x \) represents the number of bicycles Barry must sell. Using variables allows us to create general formulas and equations, making it easier to solve specific problems.

• The variable \( x \) is crucial because it is what we're solving for in Barry's income problem.
• By assigning a variable, we transform a textual problem into a mathematical expression.
• It makes it easy to manipulate and simplify the equation until we find the exact value needed.

Defining variables clearly is important at the start of any algebra problem. In this case, it helps model the entire income structure Barry relies on.

Linear Algebra

Linear algebra often involves solving equations like Barry's. In this example, we deal with a linear equation, which means the highest power of the variable is one. The problem can be addressed by simple algebraic manipulations, such as additions, subtractions, and divisions.
Here's a simple breakdown of solving Barry's linear equation:

• Isolating the variable: First, we subtract \(\$300\) from both sides. This leaves us with \(15x = 450\).
• Solving for \(x\): Finally, divide both sides by \(15\) to get \(x = 30\).

Linear equations are foundational in algebra as they can model numerous real-world scenarios. These equations provide a clear path to find solutions involving unknown quantities.

Income Calculations

Calculating income often involves understanding fixed and variable components of earnings. For Barry, his income depends on two parts:

• Fixed part: A constant monthly salary of \(\\(300\).
• Variable part: Commission based on sales, \(\\)15\) per bicycle.

To find out how many bicycles Barry needs to sell to achieve an income of \(\$750\), we consider both components. By setting up an equation where these parts sum up to the desired monthly income, we can solve for the variable representing his sales.
This kind of calculation is common in jobs where a part of the income is performance-based. It's crucial to know how to separate these components for personal finance planning and goal setting.